Optimal. Leaf size=65 \[ \frac {x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt {c x^2}}-\frac {a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt {c x^2}}-\frac {a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int x (a+b x)^n \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a x (a+b x)^{1+n}}{b^2 c (1+n) \sqrt {c x^2}}+\frac {x (a+b x)^{2+n}}{b^2 c (2+n) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.69 \[ \frac {x^3 (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 72, normalized size = 1.11 \[ \frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} c^{2} n^{2} + 3 \, b^{2} c^{2} n + 2 \, b^{2} c^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.71 \[ -\frac {\left (-x n b -b x +a \right ) x^{3} \left (b x +a \right )^{n +1}}{\left (c \,x^{2}\right )^{\frac {3}{2}} \left (n^{2}+3 n +2\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 45, normalized size = 0.69 \[ \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 80, normalized size = 1.23 \[ \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^3\,\left (n+1\right )}{c\,\left (n^2+3\,n+2\right )}-\frac {a^2\,x}{b^2\,c\,\left (n^2+3\,n+2\right )}+\frac {a\,n\,x^2}{b\,c\,\left (n^2+3\,n+2\right )}\right )}{\sqrt {c\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {a^{n} x^{5}}{2 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} x^{3} \left (a + b x\right )^{n}}{b^{2} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 3 b^{2} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 2 b^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {a b n x^{4} \left (a + b x\right )^{n}}{b^{2} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 3 b^{2} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 2 b^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} n x^{5} \left (a + b x\right )^{n}}{b^{2} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 3 b^{2} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 2 b^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{5} \left (a + b x\right )^{n}}{b^{2} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 3 b^{2} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 2 b^{2} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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